This non-crossing locking theorem has a clever proof, which can be called “bulk-cell decomposition” method or “bulk-cell gluing” method  [1]}. This intuitive proof method can be described as follows: as shown in figure REF , considering a specified set of boundary elementary regions and some non-crossing composite regions, it is easy to see that, since these regions do not cross each other, accordingly, the corresponding RT surfaces of these regions will not intersect with each other. Therefore, we can utilize this series of RT surfaces to discretize the bulk spacetime, according to some certain appropriate order. These small fragments cut by RT surfaces can be called “cells”, and each cell is bounded by RT surfaces or boundary subregions. Next, regarding each cell as an individual spacetime, by using a simpler locking theorem, which states that there is always a locking bit thread configuration can lock all elementary regions simultaneously, one can always find a locking thread configuration to lock all boundaries of each cell spacetime simultaneously. Since the boundaries of the cell spacetime are minimal surfaces, the locking thread configuration must satisfy that the threads intersecting these surfaces should be orthogonal to them and saturate the density bound. Finally, by gluing all the locking thread configurations of cell spacetimes together, one can obtain a locking thread configuration of a whole bulk spacetime, and thus prove the existence.
